**(pre-request)** We know that **time reversal operator** $T$ is an **anti-unitary operator** in Minkowsi Spacetime. i.e.

$$
T z=z^*T
$$
where the complex number $z$ becomes its complex conjugate. See, for example, Peskin and Schroeder ``An Introduction To Quantum Field Theory'' p.67 Eq (3.133).

**(Question)** Is **Euclidean time reversal operator** $T_E$ an **unitary operator**, i.e.

$$
T_E z=z T_E \;\;\;(?)
$$

or an **anti-unitary operator** in Euclidean Spacetime? Why is necessarily that or why is it necessarily not that? **Or should $T_E$ be an unitary operator like Parity $P$ instead?**

However, here see the attempt of a PRL paper, Euclidean continuation of the Dirac fermion
where time reversal $T_E$ is given by, on page 3:
$$
T_E z=z^*T_E \;\;\;(?)
$$
$T_E$ is still an **anti-unitary operator**!

It seems to me if one consider Euclidean spacetime, the Euclidean signature is the same sign, say $(-,-,-,-,\dots)$, then **Parity $P$** acts on the space is equivalent to the **Euclidean time reversal operator $T_E$** acts on Euclidean time (which is now like one of the spatial dimensions). So **shouldn't $T_E$ be an unitary operator as Parity $P$?**

[Other Refs]

i. Please, you may read, an earlier Phys.SE question here, probably is poorly formulated, so cannot draw the efforts of people to answer the question. Here let me try an easier way and focus on one issue only.

ii. Osterwalder-Schrader (OS) approach, and ``A continuous Wick rotation for spinor fields and supersymmetry in Euclidean space'' by Peter van Nieuwenhuizen, Andrew Waldron.

This post imported from StackExchange Physics at 2014-06-04 11:36 (UCT), posted by SE-user Idear